774 

 As to (14) the following theorems') obtain. 

 A. If KA■^\'è) = K{.v,^), K,{.v:i) = ^'^,. .. Kn(.v,S,)=-^^^ are 



Ox X " 



d"+iAV,£) 

 continuous and /r„ + i(A',^) = — ^ — — is tinite tor a < S, < a- < h, 



a ,u" + ' ^ ' ^ ^ 



and the discontinuities of Kn^\{x,S,), if it has any, are regularly 

 distributed ^), and if moreover K„ {cc,x) ^ 0, /Vj {:v,x) ^ 0, . . . § . . . 

 /vn — iC'i'.^O^O, but K„{x,x)~\=0 for a^x^b, (14) will have only on*? 

 continuous solution under the necessary and sufficient conditions: 

 /'(.t), ƒ'(.('),... /'^"+ '(■*') continuous for a^x^b und f{n)=^/'{ii):= . . . 

 = fi"){a}^=0. And this solution will be represented by the only 

 possible continuous solution of the integral equation of the second kind : 



ƒ(" + '■- (.,0 = a; {x.x) u {x) + i Kn+, (x4) u iS) dï, 



(16) 



B. If /v,(.f,§) = K{x,ï), K,{x,?,) = —^^,...K,,-i{x4)= ^ \*' are 



ax OA'"~i 



d"K{x,i) G(x.S,) 



continuous for n ^ï^.v^b and A'„(.r,<;)= — ^ = — {0<R{/.„)<\), 



dx" (_,._..)'„ 



dG{x,^) 



in which 6r(.7,',s) and — r are also continuous for « < E<,r<ft. and if 



ox 



moreover A o(*",'ï') ^ 0, K^{x,x) ^ 0, . . . K„—i{x,x) ^ 0, but G{x,x) =(= 

 for a ^ X ^b, — then (14) will have only one continuous solution 

 under the necessary and sufticient conditions: ƒ (.c),/'(.c) ... /'(«Vu') and 



£ C_llSïl- dS, continuous for a < x ^ b,aM f{a)=f\a)=..—f("\a)= 



a 



= 0. And this solution will be represented hy the only possible 

 continuous solution of the integral equation of the first kind: 



X 



F{x) = fL{x,i) u("s) dè, (17) 



a 



in which : 



1) See for instance M. Böcher : An introduction to the study of integral equa- 

 tions (1904) §§ 13, 14. 



2) Viz. that the discontinuities lie on a finite number of curves with continu- 

 ously turning tangents in the space of the .r^-plane considered, which are inter- 

 sected by lines // x- of $-axis always in a finite number of points. 



